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Deep Inelastic Structure Functions at two Loops

S. Moch, J. A. M. Vermaseren

TL;DR

The paper presents an analytic, NNLO-level determination of deep inelastic structure function evolution by computing Mellin moments of F2, F3, and FL up to two loops via OPE and projection methods. It develops a recursive, harmonic-sum based framework to obtain results in both Mellin space and x-space, providing complete anomalous dimensions and coefficient functions for singlet and non-singlet sectors. The x-space results, expressed with harmonic polylogarithms, enable direct phenomenological use, while the method lays groundwork for three-loop extensions and polarized structure functions. Overall, this work delivers a rigorous, cross-checked NNLO benchmark for DIS and a scalable approach toward higher-order QCD precision.

Abstract

We present the analytic calculation of the Mellin moments of the structure functions F_2, F_3 and F_L in perturbative QCD up to second order corrections and in leading twist approximation. We calculate the 2-loop contributions to the anomalous dimensions of the singlet and non-singlet operator matrix elements and the 2-loop coefficient functions of F_2, F_3 and F_L. We perform the inverse Mellin transformation analytically and find our results in agreement with earlier calculations in the literature by Zijlstra and van Neerven.

Deep Inelastic Structure Functions at two Loops

TL;DR

The paper presents an analytic, NNLO-level determination of deep inelastic structure function evolution by computing Mellin moments of F2, F3, and FL up to two loops via OPE and projection methods. It develops a recursive, harmonic-sum based framework to obtain results in both Mellin space and x-space, providing complete anomalous dimensions and coefficient functions for singlet and non-singlet sectors. The x-space results, expressed with harmonic polylogarithms, enable direct phenomenological use, while the method lays groundwork for three-loop extensions and polarized structure functions. Overall, this work delivers a rigorous, cross-checked NNLO benchmark for DIS and a scalable approach toward higher-order QCD precision.

Abstract

We present the analytic calculation of the Mellin moments of the structure functions F_2, F_3 and F_L in perturbative QCD up to second order corrections and in leading twist approximation. We calculate the 2-loop contributions to the anomalous dimensions of the singlet and non-singlet operator matrix elements and the 2-loop coefficient functions of F_2, F_3 and F_L. We perform the inverse Mellin transformation analytically and find our results in agreement with earlier calculations in the literature by Zijlstra and van Neerven.

Paper Structure

This paper contains 14 sections, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formalism
  3. Operator product expansion
  4. Renormalization
  5. Methods
  6. Reduction identities
  7. Solutions
  8. Harmonic sums
  9. Harmonic Polylogarithms
  10. Calculation and Results
  11. Renormalization and mass factorization
  12. Results in Mellin space
  13. Results in $x$-space
  14. Conclusions

Figures (1)

  • Figure 1: The basic basic topologies of a 2-point function up to two loops. Left: The basic 1-loop diagram $L1$. Right: The 2-loop topology $T1$.